基于测量的GHZ态制备的最优解码：最大效用解码器
Optimal Decoding for Measurement-Based GHZ State Preparation: The Maximum-Utility Decoder
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中文总结 AI 辅助
该研究针对基于测量的GHZ态制备中传统解码器无法最大化连续长程有序性的问题，提出最大效用解码器（MUD），构建两阶段高效算法，在N=256×256规模下缩小了87%的性能差距，为量子技术提供最优解码策略。
中文摘要 AI 辅助
宏观Greenberger-Horne-Zeilinger（GHZ）态的精细制备为计量学、密码学和容错码等量子技术提供了基础资源。尽管最先进的基于测量的协议能实现高效的低深度执行，但其性能可能受限于传统解码器，如最小权完美匹配（MWPM）甚至最大似然解码（MLD），这些解码器针对二元逻辑恢复进行优化，无法最大化二维几何中GHZ态特有的连续长程有序性。在此，我们通过将解码问题构建为最小贝叶斯风险推理来克服这一局限，提出一种最大化解码态预期效用的通用范式。实施该最大效用方法，我们构建了一种算法，实现了最高的单次解码量子序，从而为基于测量的GHZ态制备确立了最优解码策略。为提升计算效率，我们设计了可扩展的两阶段解码器，其首先将症候编码为MWPM的边权，随后用经训练以最大化预期效用的卷积神经网络优化结果，成本仅为最优解码器的一小部分。值得注意的是，仅第一阶段——该阶段使匹配能感知规范选择，且额外成本仅为基础MWPM——在我们研究的最大规模N=256×256时已接近最优性能，缩小了基础MWPM与最优解码阈值之间高达87%的差距。最大效用解码器（MUD）推广了MWPM和MLD，建立了一个通用框架，可通过重新定义效用函数明确适配特定实验的操作需求。
英文摘要
The meticulous preparation of macroscopic Greenberger-Horne-Zeilinger (GHZ) states provides a foundational resource for quantum technologies such as metrology, cryptography, and fault-tolerant codes. While state-of-the-art measurement-based protocols offer efficient low-depth execution, their performance can be bottlenecked by conventional decoders, such as minimum weight perfect matching (MWPM) or even maximum-likelihood decoding (MLD), which optimize for $binary$ logical recovery and fail to maximize the $continuous$ long-range order characteristic of a GHZ state for two-dimensional geometries. Here we overcome this limitation by framing the decoding problem as minimum Bayesian risk inference, introducing a general paradigm that maximizes the expected ${utility}$ of the decoded state. Implementing this maximum-utility approach, we construct an algorithm that achieves the highest possible per-shot decoded quantum order and thereby establish an optimal decoding strategy for measurement-based GHZ state preparation. To improve its computational efficiency, we design a scalable two-stage decoder, which first encodes the syndromes into the edge weights of MWPM and then refines the result with a convolutional neural network trained to maximize the expected utility, at a fraction of the cost of the optimal decoder. Remarkably, we find that the first stage alone$\unicode{x2014}$which makes the matching aware of the gauge choice at no cost beyond bare MWPM$\unicode{x2014}$already performs near-optimally up to the largest sizes we study, $N=256\times256$, closing up to $87\%$ of the gap between the bare-MWPM and optimal decoding thresholds. Generalizing MWPM and MLD, the maximum-utility decoder (MUD) establishes a versatile framework that can be explicitly tailored to the operational demands of specific experiments by redefining the utility function.
